Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25747 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,5)(7,8,13,9)(10,11,14,12);(3,4,6,5)(7,9,13,8)(10,11,14,12);(7,10,11,9,13,14,12,8);(3,5,6,4)92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215857, #72215961, C2×#11713628, … |
| maximal normal subgroup of | #72215857, #72215961, #72225944, … |
| maximal quotient of | #72225944, #72226238, #72226283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350498);